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Billiard table chaos wins $1 million maths prize

Cutting a hole in the pool table is unlikely to win you many fans down at your local bar, but doing the mathematical equivalent has just led to a million dollar prize.

The Norwegian Academy of Science and Letters in Oslo announced today that it has awarded Yakov Sinai of Princeton University and the Landau Institute for Theoretical Physics in Chernogolovka, Russia, its annual Abel prize.

Worth 6 million Norwegian kroner, or roughly US&dollar;1 million, it is sometimes known as the Nobel prize of mathematics. Unlike physics, chemistry and medicine, mathematics does not have a dedicated real Nobel prize. Sinai received the news at his home in New Jersey at 5 am&colon; “I usually get up very early, so it was not a problem,” he said, during a news conference this morning.

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What does all this have to do with pool tables? To understand, start with Sinai’s work, which involves studying how systems of equations change over time. Physicists use such differential equations to model all sorts of real-world problems, from planetary motion to climate change, but it isn’t always clear how well the models can predict the future.

Chaos toolbox

That is because these scenarios – and the equations that model them – behave chaotically, meaning even a small change in the starting conditions can lead to big differences in the outcome. Sinai created tools to study this chaotic behaviour that are still used today. “Much of his research has become a standard toolbox for mathematical physicists,” said mathematician Ragni Piene, chair of the Abel committee that chose the winner.

One important result looked at the motion of molecules in a gas. Real gases have millions of molecules, making them a very complex problem to study, so Sinai tried to find a simpler model whose behaviour could be fully proved. He came up with a theoretical game of billiards played on a square table with a circle removed from the middle, and replaced by a circular fence. Balls bounce off the outer square and the inner circle, just as molecules bounce off each other in a gas.

It turns out this model reproduces many of the properties of a more complex gas, including one called ergodicity, which roughly means that a ball, or gas molecule, is equally likely to reach any part of the system given enough time.

Shared language

Sinai also came up with a way to quantify chaos in many types of systems. Some systems aren’t remotely chaotic, like a particle moving along a never-changing circular path. Others appear totally chaotic because they are random, such as the numbers that come up on a repeatedly thrown dice.

In between is what mathematicians called deterministic chaos, in which the rules of a system are totally predictable, but its long-term behaviour is uncertain. Sinai, working with fellow mathematician Andrey Kolmogorov, came up with a concept called Kolmogorov-Sinai entropy, which measures the average amount of uncertainty of the system in the future.

“We often talk about mathematics in terms of the great theorems that are proved, but it’s as much about great definitions,” said Jordan Ellenberg of the University of Wisconsin in Madison, who gave a presentation about Sinai’s work following the announcement at the news conference.

Defining Kolmogorov-Sinai entropy has helped mathematicians and physicists gain a deeper understanding of chaotic systems, helping to study many real-world problems and discuss results in a shared language. “There is a saying – a good physicist is a physicist who can explain his result to mathematicians,” said Sinai.